A fully nonlinear free boundary problem arising from optimal dividend and risk control model

Focusing on the problem arising from a stochastic model of risk control and dividend optimization techniques for a financial corporation, this work considers a parabolic variational inequality with gradient constraint \begin{document}$\min\Big\{v_t-\max\limits_{0\leq a\leq1}\Big(\frac{1}{2}\sigma^2a^2v_{xx}+\mu av_x\Big)+cv,\;v_x-1\Big\} = 0.$ \end{document} Suppose the company's performance index is the total discounted expected dividends, our objective is to choose a pair of control variables so as to maximize the company's performance index, which is the solution to the above variational inequality under certain initial-boundary conditions. The main effort is to analyse the properties of the solution and two free boundaries arising from the above variational inequality, which we call dividend boundary and reinsurance boundary.


(Communicated by Juan Li)
Abstract. Focusing on the problem arising from a stochastic model of risk control and dividend optimization techniques for a financial corporation, this work considers a parabolic variational inequality with gradient constraint min vt − max 0≤a≤1 1 2 σ 2 a 2 vxx + µavx + cv, vx − 1 = 0.
Suppose the company's performance index is the total discounted expected dividends, our objective is to choose a pair of control variables so as to maximize the company's performance index, which is the solution to the above variational inequality under certain initial-boundary conditions. The main effort is to analyse the properties of the solution and two free boundaries arising from the above variational inequality, which we call dividend boundary and reinsurance boundary.

1.
Introduction. An insurance company is a typical example of a financial corporation in the problem of optimal dividend distribution and risk control, [15,16,9] considered some kinds of optimal risk and dividend control problems under some conditions. For a finite time model, we solved the problems of dividend control model and risk control model in [7] and [8], respectively. In this paper, we deal with a stochastic model of risk control and dividend optimization, which is an evolutionary problem of one of the models in [15]. [15] discussed some different types of conditions imposed upon a company and different types of reinsurance available, however, the assumption of finite time will make the problem much more complicated.
We consider that the insurance company has an option to choose the times and amounts of dividends and the reinsurance policies before T ∧ τ , here T is the terminal time and τ is the bankruptcy time. The type of reinsurance in this paper is proportional reinsurance, that is, it requires the reinsurer to cover a fraction of each claim equal to the fraction of total premiums he receives from the cedent.
In recent years, there is growing interest in applying parabolic variational inequality with gradient constraint and Barenblatt equation in mathematical finance, see for example [2,1,3,10,11,17]. However, up to date, there seems to be no results on parabolic Barenblatt equation with gradient constraint. Our primary motivation stems from the free boundaries arising from the Barenblatt equation with gradient constraint.
Specifically, we focus on the following problem Since v x ≥ 1, the maxima of a in [0, 1] is a * (x, t) = arg max if v xx ≥ 0.
In the case − µ σ 2 vx vxx < 1 and v xx <0, then a * < 1, which means that the insurance company shall transfer 1 − a * = 1 + µ σ 2 vx vxx ratio of incomes to reinsurance company, the corresponding region denoted by R, called reinsurance region. In the other case − µ σ 2 vx vxx ≥ 1 or v xx ≥ 0, then a * = 1, which means the insurance company is not need to transfer any risk to the reinsurance company, the corresponding region denoted by N R. That is Therefore, in order to prove the existence of solution and its regularity, we should use the method of approaching to construct non-degenerate approaching problem whose v x , v xx are nonsingular on x = 0. However, the proof of convergence is not easy, because we need to verify the uniform parabolic condition for the approaching problem holds in the interior of the domain, which is highly technical. Meanwhile, we can see the free boundary between R and N R (Reinsurance free boundary) is the set In order to obtain the existence and the smoothness of this free boundary line, the directly perception is to discuss the monotonicity and the smoothness of the function − µ σ 2 vx vxx or v xx + µ σ 2 v x , but this is not an easy task. To proceed, we deal with it in another way, observing from the equations (3) and (4) we found that the function has the same signs with v xx + µ σ 2 v x whether in R or N R and it is more convenient to analyze. Therefore, we turn to W (x, t) to research the reinsurance free boundary line. This is an innovation point of this article.
The rest of the paper is arranged as follows. The precise formulation of the problem and the HJB equation are given in Section 2. Section 3 shows the existence and uniqueness of the solution to problem (1) and obtains some properties of the solution. Section 4 characterizes the properties of the dividend boundary h(t). Section 5 shows the existence and uniqueness of the reinsurance boundary g(t), moreover, it is a differentiable curve and 0 < g(t) < h(t). We present the optimal strategy to the financial model in section 6. In Appendix A, we give the verification theorem to show that the solution to the HJB equation coincides with the value function. In Appendix B, we present the expression of solution to corresponding steady problem. In Appendix C, we show the proof of Lemma 5.1. In Appendix D, we confirm value function (7) is a viscosity solution of (9).
2. Formulation. In this paper, we work with a complete filtered probability space (Ω, F, {F t } t≥0 , P), where (Ω, F, P) is the probability space, and {F t } t≥0 is the filtration satisfying the usual condition. We consider that a financial corporation has an option to choose the policies of the dividends distribution and the reinsurance part of the claims (the cedent is required to pay simultaneously diverting part of the premiums to a reinsurance company). Let R t be the dynamics of the reserve process, it satisfies where µ, σ are positive constants and W s is a standard Brownian motion, control process (a s , L s ) is adapted to F s , 0 ≤ 1 − a s ≤ 1 corresponds to the reinsurance fraction, and L s ≥ 0 is a non-decreasing, càdlàg process, representing the cumulative dividends up to time s. We denote by Π t = {(a s , L s )|s ≥ t} the set of all admissible control processes from time t with R t− = x. Our objective is to choose an optimal control process so as to maximize the company's discounted expected dividends up to time T ∧ τ , where τ is the bankruptcy time, defined as τ = inf{s ≥ t : R s ≤ 0}. (6) With the control process (a, L), we can define the company's performance index as And the value function is Using the theory of viscosity solution in differential equations, one can obtain the following HJB equation On the other hand, when x = 0, then τ = t and L t − L t− = 0, thus V (0, t) = 0.
3. Solvability of problem (9). For the convenience in the analysis, we analyze the backward equation (9) by studying its associated forward equation of the following form. Let v(x, t) = V (x, T − t) in (9), then v(x, t) satisfies We will show that there is a constant x d > 0(the dividend free boundary point of corresponding steady problem) such that v x = 1 when x ≥ x d . So we only need to consider the above problem in the truncated domain, where The aim of this section is to prove the existence of W 2,1 p solution to problem (10) and some useful estimates. The main results are shown in Theorem 3.8. Firstly, we rewrite (10) to a fully non-linear problem (11). And then we construct the penalty approximation problem (14). In order to get the solution to problem (14), we turn to problem (20), whose solution is the gradient of solution to problem (14). After discussing problem (20). we succeed in proving Theorem 3.8.
3.1. The equivalent problem (11). Note that (10) is a Barenblatt equation and it is not convenience to research it directly, we should change it to a specific fully non-linear equation. Define a function A(z) as It is not hard to see that and A (±∞) := lim z→±∞ A (z) = 0. Therefore, (10) can be rewritten as Define and Since which we will repeatedly use later.
In particular, this fact is sufficient to establish the comparison principle associated to the operator L and F.

3.2.
The penalty approximation problem (14) and (20). In order to prove the existence of a solution to problem (11), we construct the following penalty approximation problem. where and penalty function β ε (t) (see Figure 1) satisfies where Owing to the boundary condition on x = 0 and the equation in (14), we have We claim that for any ε small enough, the above boundary condition is equivalent to the following condition In fact, according to the definition of A(z), There is a contradiction in (19), which implies only (18) holds true. Since we could deduce that the solution of (14) satisfies v ε x ≥ 1 and v ε xx ≤ 0(see (36)), together with (18), we obtain the boundary condition (17). Denote 3.3. The solution to problem (20). Note that the boundary condition and initial condition in (20) are not consistent at (0, 0). In order to have existence of classical solution, we construct the following approximation problem where f δ (t) ∈ C 2 ([0, +∞)) and Since the boundary condition in (21) is unusual, we should establish the comparison principle associated to (21).
x to obtain This equation is in divergence form of u ε,δ x with bounded coefficients. Moreover, note that together with u ε,δ ≥ 1 and f δ ≤ 1 implies From u ε,δ ≥ 1 and u ε,δ (N, t) = 1 we have Combining with by the maximum principle for weak solution(see [12]), we deduce that u ε,δ x ≤ 0.
where u ε is a solution to problem (20). Then Integrating the equation in (20) from 0 to x, we get v ε satisfies the equation in (14). In addition, it is easy to verify that v ε satisfies the initial and boundary conditions in (14).
3.5. Solvability of problem (11). In order to obtain uniform interior estimates to u ε independent of ε, we need uniform parabolic conditions associated to L ε and F ε inner the domain, which claims that for uniform positive constants λ and Λ. Thanks to A(z) ≤ 1, we only need < N ), there exists κ > 0 which is independent of ε (but depends on a), such that Since u ε ≥ 1, it is only needed to prove Proof. Fix t ∈ [0, T ], mean value theorem claims that there exists a ξ ∈ ( a 2 , a) such that where, the second inequality is due to the estimate (34). Owing to u ε xx ≥ 0, then Theorem 3.8. Problem (11) has a unique solution v ∈ W 2,1 p, loc (Q T ) C(Q T ), and v x ∈ W 2,1 p, loc (Q T ) C(Q T ∪ {t = 0}), moreover, we have the following estimates where, γ = 1 − η, K, p, η are defined as (25).

Rewrite (28) as
We emphasize that the above C are independent of ε. Let ε → 0, u ε lead to a limit u ∈ W 2,1 p, loc (Q T ) (probably a subsequence) in the fixed domain Q a in the sense u ε → u weakly in W 2,1 p (Q a ). Moreover the Sobolev embedding theorem implies that u ε → u in C 1+α, 1+α 2 (Q a ). Apply W 2,1 p interior estimate to (14) and we deduce that Letting ε → 0, we also have a subsequence of v ε lead to a limit v ∈ W 2,1 In the following, we show that v satisfies the variational inequality in (11). Firstly, by (34) we have v x = u ≥ 1. By the equation in (14) let ε → 0 and we get v t − Lv ≥ 0. If there exists a point (x, t) such that v x (x, t) > 1, then there exists ε 0 such that for all ε < ε 0 , v ε x (y, t) ≥ v ε x (x, t) > 1 + ε 0 , y ≤ x, then the right hand side of (46) is 0 at (x, t), send ε approach to 0, we have So v satisfies the variational inequality in (11).
Moreover, (42)-(45) follow from (34)-(37). Integrating both sides of inequality (42), (43), we obtain (40), (41). Now we will prove the uniqueness. Suppose v 1 , v 2 are two solutions of (11). Set Comparison principle for fully nonlinear equation (see [4], red p.52, Theorem 16) 4. Properties of the dividend boundary. In this section, we get existence of the dividend free boundary to the variational problem (10)(Theorem 4.1). We also prove that it is a strictly increasing C ∞ smooth curve and obtain its upper bound and the location of starting point(Theorem 4.2, Theorem 4.4). In addition, based on Theorem 3.8, we further prove the solution to (10) belongs to C 2,1 (Q T )(Theorem 4.3), which is important to the verification theorem. Now, in order to characterize the optimal dividend boundary, we define Moreover, h(t) is increasing. Proof.
Since v x is decreasing w.r.t. x, we can define It is obviously that (47) holds true. The monotonicity easily follows from the fact v xt ≥ 0.
In order to characterize the properties of h(t), we focus on the neighborhood of (h(t), t). Note that v x ∈ W 2,1 p, loc (Q T ), Sobolev embedding theorem implies that v By the definition of D and (47), we know Thus Similar to the proof of Theorem 3.8, let ε → 0 in (20), we get u = v x satisfying Theorem 4.3. The solution to problem (11) v xt ∈ C(Q T ), moreover, v ∈ C 2,1 (Q T ).
Proof. Since u = v x satisfies (52) in the neighborhood of (h(t), t), using the method in [5] we get v xt ∈ C(Q T ), together with (50), it is obvious that v ∈ C 2,1 (Q T ).

Theorem 4.4. The free boundary h(t) is strictly increasing with
where x d is defined in (73), moreover, h(t) ∈ C ∞ (0, T ].
Proof. We first prove the strictly monotonicity. If not, there exists t 1 , , we have u = 1 and u x = 0 on Γ, thus u t = 0, u xt = 0 on Γ. Suppose B is the neighborhood where (52) holds, applying Hopf lemma(see [4]) or u xt < 0 on Γ, but both come to contradictions. Now we will prove (53). Consider the following steady variational inequality problem It is the HJB equation to the infinite time model and the expression of solution V ∞ (x) and free boundary points x d and x r were obtained by [15]. We will present them in Appendix B. In particular, we give In the following, we show that From the equation in (20), let ε → 0, u = v x satisfies the following variational inequality In order to prove (55), we have to give a comparison principle for the above variational inequality: Suppose u 1 , u 2 ∈ C 2,1 (Q T ) C(Q T ) satisfy (56) with the following initial and boundary conditions The comparison principle claims . In order to prove (55), we have to show that for any a > 0, there exists u a ≤ U b for small enough b > 0. It is easy to verify that both u a and U b satisfy the variational inequality (56) with the following boundary and initial conditions on Hence, by the comparison principle, we have . Since a is arbitrary, (55) holds, which implies h(t) ≤ x d .

5.
Properties of the reinsurance boundary. In this section, we obtain the existence of reinsurance free boundary, and prove that it is a C 1 smooth curve. (Theorem 5.3) Firstly, we define Notice that D ⊂ {v xx = 0} ⊂ N R, then R ⊂ N D, which means that the optimal reinsurance free boundary locates in N D.
We hope to find a function W (x, t) to characterize the optimal reinsurance free boundary, specifically, we hope to find a function satisfying and when − µ σ 2 vx vxx < 1, i.e., A < 1, Thus W (x, t) satisfies (57) and (58). In addition, in view of the equation in (11), we have Thanks to (40)-(42), we get therefore, we can define the reinsurance free boundary as To determine the position of g(t), we need the following lemma. Proof. It seems difficult to prove from (11) by the method of PDE, but since we've got the solution to problem (11) belongs to C 2,1 (Q T ) C(Q T ) in Theorem 4.3, we could apply verification theorem (which is presented in Appendix A) to prove that V (x, t) = v(x, T − t) is the value function defined by (7). So we turn to prove that by the techniques of stochastic analysis, this work is given in Appendix C.
Lemma 5.2. The solution v to the problem (11) satisfies v t ≤ e −ct µ.
Proof. Differentiate the equation in (14) w.r.t. t , we have Denote the right hand side of the above equation as F ε (x, t), thanks to (43), we know F ε (x, t) ≤ 0. In view of the boundary conditions in (14) Construct the following approximation problem, where f δ (t) ∈ C 2 ([0, T ]) is given in (22) and satisfies Thanks to the Hölder continuity of F ε (x, t) and A ε (x, t), (63) has a unique solution w ε,δ ∈ C 2+α,1+ α 2 (Q T ), in particular, the comparison principle implies w ε,δ ≤ e −ct µ. Use the technique in the proof of Theorem 3.4, we could show that when δ → 0, w ε,δ converge to a function w ε ∈ C 2+α,1+ α 2 (Q T \ {(0, 0)}), which is the solution to (62). By the uniqueness of the solution to (62), we have w ε = v ε t , thus , v ε t ≤ e −ct µ. Letting ε → 0, we get the conclusion. Now we are able to characterize the free boundary g(t) as follows.
By Schauder interior estimate, v ∈ C 2+α,1+ α 2 (N D). Differentiate both sides of the above equation w.r.t. x and t, respectively, we get Similarly, we can deduce from Schauder interior estimate that v which implies g(t) ∈ C 1 (0, T ]. Figure 3. Reinsurance free boundary At the end of this section, we show that problem (11) degenerates on the left boundary x = 0. To see that, we give Proof. From g(t) > 0 for all t ∈ (0, T ], we know v satisfies (3) near x = 0. Applying Lemma 5.1 and Lemma 5.
6. The optimal strategy. In this section, we will give an optimal strategy to the original financial model. Set Choose the following dividend policy L * s as That is, when R * s− > h T (s), the insurance company has to pay dividend R * s− −h T (s) at time s, when R * s− < h T (s), the optimal strategy is non-dividend. Meanwhile, take Choose the following risk control strategy, We will prove that under strategy (a * , L * ), the company can get the optimal expected discounted cumulative dividends up to terminal date. The proof is presented in Appendix A.
Appendix A. . The following theorem shows that (a * , L * ) defined by (67)-(69) is the optimal strategy, and the solution of problem (9) V (x, t) is the value function defined by (7).
Theorem A.1. Suppose L * s satisfies (67) and (68), a * s satisfies (69), and V ∈ C 2,1 (Q T ) ∩ C(Q T ) is the solution to the problem (9), then for any (a s , L s ) ∈ Π s , Moreover, Proof. We first prove (70). For any (a s , L s ) ∈ Π s , assume R s is the solution to (5) with the control variables (a s , L s ), τ is the bankruptcy time of R s defined in (6), by Itô formula, where, L c s is continuous part of L s . The first two terms are non-negative. Meanwhile, we can deduce from the fact V x ≥ 1 that which is (70).
In the following, we will prove (71). Assume R * s is the solution to problem (5) with control variables (a * s , L * s ) defined in (67)-(69), τ * is the corresponding bankruptcy time, Itô formula gives So (71) holds.
Appendix B. . In this section, we introduce the solution of (54), which is the value function of infinite-time model. By the literature of [15], the solution of (54) belongs to C 2 (0, ∞) and it is an increasing concave function which can be expressed as: where θ 1 , θ 2 (θ 1 < 0 < θ 2 ) are two roots of We emphasize that x r , x d are the corresponding reinsurance boundary and dividend boundary to (54), respectively. The optimal strategy a * can be expressed as a function of x a * (x) = x x r ∧ 1.
Appendix C. . Now we prove the function V defined by (7) satisfies Define a k,m (x) := min{kx, m}, k > 0, m > 0. Suppose R k,m s is the solution to the following stochastic differential equation, with L s = 0, t ≤ s < T, R k,m T , s = T. Then when R k,m s < m, s < T , R k,m s is a geometric Brownian motion, so bankruptcy will not happen, thus It is easy to deduce that for any λ > 0, By the definition of the value function V , for any k > 0 On the other hand, for fix k, a k,m (x) satisfy the uniform Lipschitz condition, i.e. for any x 1 , x 2 > 0, and lim m→+∞ a k,m (x) = kx. Thus, when m → ∞, the solution of (75) converges to the following geometric Brownian motion in the sense of L 2 In particular, the solution to the above SDE is Thanks to (77) and (76), we have By the mean value theorem and the monotonically of V x (x, t), we prove (74).
Appendix D. . In the following, we confirm value function (7) is a viscosity solution of (9). To this end, we need to use the principle of dynamic programming(see [13]), i.e., for any stopping time V (y, s).
Now we prove viscosity supersolution property. Let (x, t) ∈ (0, +∞) × [0, T ) and let ϕ ∈ C 2 ((0, +∞) × [0, T )) be a test function such that By definition of V * (x, t), there exists a sequence (x m , t m ) in (0, +∞) × [0, T ) such that By the continuity of ϕ and by (79) we also have that For any fix admissible reinsurance strategy a, choose L s ≡ 0 for s ≥ t m , we denote by R tm,xm s the associated controlled process. Let after noting that the stochastic integral term cancels out by taking expectations since the integrand is bounded. By a.s. continuity of the trajectory R tm,xm s (here L s ≡ 0), it follows that for sufficiently large m(≥ N (ω)) ,τ m (ω) ≥ t m + h m , i.e., θ m (ω) = t m +h m a.s.. Thus, by the mean value theorem, the random variable inside the expectation in (82) converges a.s. to − ϕ t − 1 2 σ 2 a 2 ϕ xx − µaϕ x + cϕ (x, t) when m goes to infinity. Moreover, this random variable is bounded by a constant independent of m, so applying dominated convergence theorem, let m goes to infinity we obtain − ϕ t − 1 2 σ 2 a 2 ϕ xx − µaϕ x + cϕ (x, t) ≥ 0.